Continuity of a multivariable function

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Homework Help Overview

The problem involves determining the continuity of a multivariable function defined piecewise, specifically examining the behavior of the function as it approaches a certain point in the context of limits.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss various methods for evaluating the limit as (x,y) approaches (2,0), including substitution and the use of different paths. There is mention of encountering indeterminate forms and the exploration of elliptical coordinates as a potential approach.

Discussion Status

The discussion is active, with participants sharing different strategies and questioning the applicability of certain methods. Some guidance has been offered regarding the use of elliptical coordinates, but there is no consensus on the best approach yet.

Contextual Notes

One participant expresses uncertainty about using elliptical coordinates due to a lack of familiarity, indicating a potential gap in knowledge that may affect their ability to engage with the suggested methods.

Jalo
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Homework Statement



Given the function:

x*y / (4-x²-2y²) if x²+2y² ≠4
0 if x²+2y² = 4

Check if the function is continuous.



Homework Equations





The Attempt at a Solution



I tried using various ways to see if the result of the limit as (x,y)→(2,0) was the same, such as y=x-2, y=(x-2)², etc..
I didn't manage to prove that the limit didn't existed. I always arrive at the 0/0 indetermination...

If anyone could point me in the right direction I'd appreciate!

D.
 
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convert to ellyptic coordinates x=2rcos(t); y=sqt(2)rsin(t) and compute the limit as r tends to 0,using L'hopital's rule.If the limit is not identically 0 (or depends on t),then the function is not continuous at the poins on the ellipse.
 
hedipaldi said:
convert to ellyptic coordinates x=2rcos(t); y=sqt(2)rsin(t) and compute the limit as r tends to 0,using L'hopital's rule.If the limit is not identically 0 (or depends on t),then the function is not continuous at the poins on the ellipse.

Hmm I'll try doing it. However I never learned ellyptic coordinates. I'm wondering if there's any other way to solve this!
 
Last edited:
o.k don't name it,just substitute in the function and compute the limit as r tends to 1 and t is constant.
 

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